On qualitative multi-attribute group decision making and its consensus measure: A probability based perspective

This paper focuses on qualitative multi-attribute group decision making (MAGDM) with linguistic information in terms of single linguistic terms and/or flexible linguistic expressions. To do so, we propose a new linguistic decision rule based on the concepts of random preference and stochastic dominance, by a probability based interpretation of weight information. The importance weights and the concept of fuzzy majority are incorporated into both the multi-attribute and collective decision rule by the so-called weighted ordered weighted averaging operator with the input parameters expressed as probability distributions over a linguistic term set. Moreover, a probability based method is proposed to measure the consensus degree between individual and collective overall random preferences based on the concept of stochastic dominance, which also takes both the importance weights and the fuzzy majority into account. As such, our proposed approaches are based on the ordinal semantics of linguistic terms and voting statistics. By this, on one hand, the strict constraint of the uniform linguistic term set in linguistic decision making can be released; on the other hand, the difference and variation of individual opinions can be captured. The proposed approaches can deal with qualitative MAGDM with single linguistic terms and flexible linguistic expressions. Two application examples taken from the literature are used to illuminate the proposed techniques by comparisons with existing studies. The results show that our proposed approaches are comparable with existing studies.     

属性值为区间数的多属性决策对象排序研究

多属性决策问题的实质利用已有决策信息,通过一定方式对备选方案进行排序和择优。针对属性值为区间数的多属性决策对象排序问题,首先,提出决策对象优势关系这一概念,再次,得出决策对象的优势关系与其属性值两端点的实数值之和存在等价关系;最后,利用属性值为区间数的多属性决策实例对该算法进行了实例分析,并利用优势关系对决策对象进行排序并择优。     

基于限制优势关系的粗糙决策分析模型

针对信息不完全的偏好多属性决策问题,给出一种基于拓展粗糙集的决策分析模型.首先提出限制优势关系的概念;然后在限制优势关系下得到知识的粗糙近似,给出分类决策规则.对比分析证明,限制优势关系既保留了扩展优势关系的优点,又在一定程度上克服了扩展优势关系的局限性.最后通过一个实例验证了所提出的模型对信息不完备决策系统的处理更符合实际情况。     

Estimating Conditional Probabilities of Terrorist Attacks: Modeling Adversaries with Uncertain Value Tradeoffs

Many analyses conducted to inform security decisions depend on estimates of the conditional probabilities of different attack alternatives. These probabilities are difficult to estimate since analysts have limited access to the adversary and limited knowledge of the adversary’s utility function, so subject matter experts often provide the estimates through direct elicitation. In this article, we describe a method of using uncertainty in utility function value tradeoffs to model the adversary’s decision process and solve for the conditional probabilities of different attacks in closed form. The conditional probabilities are suitable to be used as inputs to probabilistic risk assessments and other decision support techniques. The process we describe is an extension of value‐focused thinking and is broadly applicable, including in general business decision making. We demonstrate the use of this technique with simple examples.     

Risk Decision Making in Operational Safety Management– Experience from the Nordic Benchmark Study

Technical Research Centre of Finland (VTT) and Studsvik AB, Sweden, have simulated decision making of the Swedish Nuclear Power Inspectorate and a power company by applying decision models in a benchmark study. Based on the experience from the benchmark study, a decision analysis framework to be used in safety related problems is outlined. By this framework both the power companies and the safety authorities could be provided with a more rigorous, systematic approach in their decision making. A decision analytic approach provides a structure for identifying the information requirements of the problem solving. Thus it could serve as a discussion forum between the authorities and the utilities. In this context, probabilistic safety assessment (PSA) has a crucial role of expressing the plant safety status in terms of reactor core damage accident probability and of risk contributions from various accident precursors. However, a decision under uncertainty should not be based solely on probabilities, particularly when the event in question is a rare one and its probability of occurrence is estimated by means of different kinds of approximations.     

Probabilistic dominance criteria for comparing uncertain alternatives: A tutorial

This paper is a tutorial which demonstrates the current state-of-the-art methods for incorporating risk into project selection decision making. The projects under consideration might be R&D, IT, or other capital expenditure programs. We will show six decision making methods: 1. mean-variance (MV), 2. mean-semivariance, 3. mean-critical probability, 4. stochastic dominance, 5. almost stochastic dominance (ASD), and 6. mean-Gini. We will also describe the assumptions about the risk attitudes of the decision maker which are associated with each of the techniques. While all these methods have been previously applied elsewhere, this is the first paper which shows all of their applications in the project selection context, together with their interrelationships, strengths and weaknesses. We have applied all six techniques to the same group of five hypothetical projects and evaluated the resulting nondominated sets. Among the methods reviewed here, stochastic dominance is recommended because it requires the least restrictive assumptions. ASD and mean-Gini are recommended when stochastic dominance is not practical or when it does not yield definitive choices. MV, mean-semivariance, and mean-critical probability are shown to be flawed.     

Decision making with stochastic dominance: An expository review

This paper surveys the use of stochastic dominance to decision making under uncertainty. The first part presents the relevant definitions and some properties of distributions satisfying one of the stochastic dominance conditions. These properties include restrictions on moments, an invariance property, and properties of random variables related by an exact formula.The second part contains some applications of the stochastic dominance method and especially the problem of selecting optimal portfolios. Most of the results in this section deal with conditions that make diversification an optimal strategy.     

Portfolio Selection Using Stochastic Dominance Criteria

The direct application of stochastic dominance criteria to portfolio selection problems has been thought impractical because an extremely large number of combinations of returns must be considered. This paper proposes and evaluates a rigorous statistical procedure for sampling the combinations of returns on candidate risky assets so that stochastic dominance criteria may be used directly in an efficient linear programming model for portfolio selection. The sampling scheme exploits the association of the return on each candidate stock with the return on a market index in a manner analogous to the Sharpe single-index model, thereby eliminating the large number of combinations with probabilities close to or equalling zero. Portfolios computed by the proposed linear programming stochastic dominance model are compared with those computed by the single-index quadratic programming model, using 180 months of recent data on a sample of NYSE common stocks.     

基于一致性度量的概率模糊语言多属性群决策方法

人们通常习惯用语言术语来表达他们的偏好，因此概率型语言术语集（Probabilistic Linguistic Term Set，PLTS）在决策过程中有着十分重要的作用。目前PLTS的研究刚刚起步，有关PLTS的相关研究没有关注到一致性度量的问题，对于PLTS的多属性群决策方法有待进一步研究。首先，给出了一种新的PLTS的集结方法，并且在已有的PLTS可能度公式的基础上，构建了PLTS的相似度量方法，在此基础上，进一步提出了基于PLTS一致性度量的多属性群决策方法。该方法在各决策者权重未知的情况下，考虑到各决策者之间的一致性。首先，定义PLTS的一致性度量公式，确定决策者权重；并根据PLTS的集结方法，集结各决策者的评价信息；最后，利用可能度公式对PLTS进行排序。通过案例分析验证了该方法的可行性和优越性。     

Measuring Subjective Probabilities: The Effect of Response Mode on the Use of Focal Responses,Validity, and Respondents’ Evaluations

Subjective probabilities are central to risk assessment, decision making, and risk communication efforts. Surveys measuring probability judgments have traditionally used open‐ended response modes, asking participants to generate a response between 0% and 100%. A typical finding is the seemingly excessive use of 50%, perhaps as an expression of “I don't know.” In an online survey with a nationally representative sample of the Dutch population, we examined the effect of response modes on the use of 50% and other focal responses, predictive validity, and respondents’ survey evaluations. Respondents assessed the probability of dying, getting the flu, and experiencing other health‐related events. They were randomly assigned to a traditional open‐ended response mode, a visual linear scale ranging from 0% to 100%, or a version of that visual linear scale on which a magnifier emerged after clicking on it. We found that, compared to the open‐ended response mode, the visual linear and magnifier scale each reduced the use of 50%, 0%, and 100% responses, especially among respondents with low numeracy. Responses given with each response mode were valid, in terms of significant correlations with health behavior and outcomes. Where differences emerged, the visual scales seemed to have slightly better validity than the open‐ended response mode. Both high‐numerate and low‐numerate respondents’ evaluations of the surveys were highest for the visual linear scale. Our results have implications for subjective probability elicitation and survey design.     

Using Expert Judgments to Explore Robust Alternatives for Forest Management under Climate Change

We develop and apply a judgment‐based approach to selecting robust alternatives, which are defined here as reasonably likely to achieve objectives, over a range of uncertainties. The intent is to develop an approach that is more practical in terms of data and analysis requirements than current approaches, informed by the literature and experience with probability elicitation and judgmental forecasting. The context involves decisions about managing forest lands that have been severely affected by mountain pine beetles in British Columbia, a pest infestation that is climate‐exacerbated. A forest management decision was developed as the basis for the context, objectives, and alternatives for land management actions, to frame and condition the judgments. A wide range of climate forecasts, taken to represent the 10–90% levels on cumulative distributions for future climate, were developed to condition judgments. An elicitation instrument was developed, tested, and revised to serve as the basis for eliciting probabilistic three‐point distributions regarding the performance of selected alternatives, over a set of relevant objectives, in the short and long term. The elicitations were conducted in a workshop comprising 14 regional forest management specialists. We employed the concept of stochastic dominance to help identify robust alternatives. We used extensive sensitivity analysis to explore the patterns in the judgments, and also considered the preferred alternatives for each individual expert. The results show that two alternatives that are more flexible than the current policies are judged more likely to perform better than the current alternatives on average in terms of stochastic dominance. The results suggest judgmental approaches to robust decision making deserve greater attention and testing.     

When Does Information about Probability Count in Choices Under Risk?

The question addressed in the present research is whether in naturalistic risky decision environments people are sensitive to information about the probability parameter. In Study 1, we showed that in naturalistic scenarios participants generally revealed little interest in obtaining information about the outcomes and probabilities. Moreover, participants asked fewer questions about probabilities for scenarios containing moral considerations. In Study 2, it was shown that, when supplied with information on probabilities, people could be sensitive to this information. This sensitivity depends on two factors. People were less sensitive to probabilities in scenarios perceived as containing ethical considerations. People were also less sensitive to probabilities when they were faced with a single-choice situation than when they were faced with a series of lotteries with different probabilities. This can be accounted for in terms of the evaluability principle.     

权重信息未知情况下的多属性群决策方法及其拓展

本文针对群决策中专家权重及指标权重难以确定的问题,提出一种在权重信息完全未知情况下的基于证据距离和模糊熵权变换的多属性群决策方法,其核心在于如何仅通过决策矩阵客观地确定决策者权重及指标权重。通过信息熵和证据距离确定专家权重,并利用模糊变换原理,将专家权重向量与指标熵权矩阵合成,得到统一的群体决策指标权重;最后使用线性加权法集成所有专家对备选方案的评价信息,得到整个方案集的排序。实验结果及相关讨论表明,该方法概念清晰,计算量适中,具有较强的客观性,而且易于机器实现,是一种可行、有效的多属性群决策方法。最后将该方法推广到属性值由精确数、语言值、区间数、直觉模糊数等多种形式构成的混合型多属性群决策中。     

Evaluating the administrative efficiency of courts

In addition to being held accountable for judicial decision, courts, like other public agencies, can and should be evaluated in terms of their administrative efficiency. This paper illustrates how courts can be evaluated in terms of their relative administrative efficiency, using a new approach—data envelopment analysis (DEA)—first proposed by Charnes et al. [1]. The DEA is based upon the economic notion of Pareto optimality which states that a given decision making unit (DMU) is inefficient if some other DMU, or some combination of other DMUs, can produce at least the same amounts of all outputs with less of some resource input and not more of any other resource. Conversely a DMU is said to be efficient if the above is not possible. Charnes et al. [1] generalized the usual input/output ratio measure of efficiency for a given unit in terms of a fractional linear program with fractional constraints. In the case of courts, the efficiency of any particular court is calculated by forming the ratio of a weighted sum of outputs to a weighted sum of inputs, where the weights for both outputs and inputs are to be selected in a manner that calculates the Pareto-Koopmans efficiency of the court. This paper reviews the DEA method and illustrates its application to a data base for 100 criminal superior courts in North Carolina.     

多标准决策表中发现概率规则的变精度粗糙集方法

用优势关系代替不可分辨关系,本文提出了一种可以从多标准决策表中获取概率规则的扩展变精度粗糙集模型,该模型能够处理多标准决策表中可能的不一相容性,获取由偏好对象组成的概率决策规则集,并导出事例决策系统的偏好模型。研究结果表明:基于优势关系,从多标准决策表中获取的最小概率规则集,使用的条件数量较少,且导出规则的数量较少、较强。     

基于概率权重函数和随机占优准则的开放式基金评级

传统的开放式基金评级方法存在两个缺陷,首先是忽视了现实中投资者是如何做决策的,假定投资者对利益和损失的主观态度相同;其次是忽略了现实的样本性质,假定随机收益的样本达到渐近正态的规模。通过期望效用的高阶泰勒序列展开建立超额收益的高阶矩和效用函数的关系,以高阶矩为约束条件估计样本的经验概率,再对经验概率进行决策权重调整。在此基础上,通过扩展夏普比和应用随机占优准则进行基金评级,并对645种开放式基金进行了案例分析。     

基于Heronian平均算子的Pythagorean模糊不确定语言决策方法

针对评价信息为Pythagorean模糊不确定语言数、属性权重未知且属性间相互影响的多属性决策问题,提出一种基于灰色关联法和Heronian平均算子的决策方法。首先,提出Pythagorean模糊不确定语言Heronian平均（PFULHM）算子和Pythagorean模糊不确定语言几何Heronian平均（PFULGHM）算子,并证明其满足幂等性、单调性、有界性及可交换性。考虑到属性权重之间重要程度的差异,定义了Pythagorean模糊不确定语言加权Heronian平均（PFULWHM）算子和Pythagorean模糊不确定语言加权几何Heronian平均（PFULWGHM）算子。其次,将灰色关联法运用到Pythagorean模糊不确定语言环境中求解属性权重。最后,提出基于PFULWHM算子和PFULWGHM算子的决策方法,并通过算例分析说明本文提出方法的有效性。     

群体两两比较判断集成的最短距离方法

本文将Yu和Cook等提出的一般距离的概念应用到群体决策问题中,基于AHP原理,给出了群体决策集成问题的定义,综述了现有的两两比较判断集成方法,提出了一种基于最短距离的集成方法——最短距离方法(Mini-mum Distance Method,MSM),并用数值例示进行了说明,最后讨论了群体决策以及AHP未来可能的研究课题。     

Measuring uncertainty

The decision analysis approach to problem solving is widely documented. A range of articles and books have considered ways for analysing problems under uncertainty and also methods for formalising an individual's attitude towards risk in terms of cardinal utility measures. In order to carry out such analyses the uncertainty in any problem has to be explicitly quantified in probabilistic terms. The present article reviews procedures for assessing probabilities and proposes practical guidelines indicating those methods which, in the authors' opinion, are most fruitful for executives to use.